# Some Stratospherically Strenuous Sudoku

These three sudoku are particularly irksome, can you logically deduce the solutions to all (or any!) ...and provide your reasoning?

They are all proper sudoku - each has a unique solution.

They can, of course, be solved by brute force (I can get their solution in milliseconds too).

1. "$13$ cigars, $8$ doughnuts, and $2$ false exits for Columbo."

   1  2  3   4  5  6   7  8  9
+---------+---------+---------+
A| 1  ·  · | ·  9  · | ·  ·  5 |A
B| ·  ·  4 | ·  ·  · | 1  ·  · |B
C| ·  7  · | ·  ·  1 | ·  6  · |C
+---------+---------+---------+
D| ·  ·  9 | ·  3  · | 2  ·  · |D
E| ·  ·  · | ·  ·  8 | ·  ·  3 |E
F| 3  ·  · | 9  ·  5 | ·  ·  · |F
+---------+---------+---------+
G| 2  ·  · | ·  8  · | ·  ·  9 |G
H| ·  ·  1 | ·  ·  · | 7  ·  · |H
J| ·  6  · | ·  ·  · | ·  4  · |J
+---------+---------+---------+
1  2  3   4  5  6   7  8  9

2. "$59$ little grey cells for Poirot."

       1  2  3   4  5  6   7  8  9
+---------+---------+---------+
A| ·  ·  9 | ·  ·  3 | 4  ·  · |A
B| ·  2  · | ·  ·  · | ·  ·  8 |B
C| 1  ·  · | ·  ·  · | ·  5  · |C
+---------+---------+---------+
D| ·  8  · | ·  ·  · | ·  ·  1 |D
E| 5  ·  · | ·  ·  · | ·  2  · |E
F| ·  ·  6 | ·  ·  7 | 9  ·  · |F
+---------+---------+---------+
G| ·  ·  · | ·  3  5 | ·  ·  · |G
H| ·  ·  3 | 6  7  · | ·  ·  · |H
J| ·  ·  · | 9  4  · | 7  ·  · |J
+---------+---------+---------+
1  2  3   4  5  6   7  8  9

3. "$22$ premises for Sherlock."

   1  2  3   4  5  6   7  8  9
+---------+---------+---------+
A| 3  ·  · | ·  ·  6 | ·  4  · |A
B| ·  ·  · | ·  2  · | 1  ·  · |B
C| ·  5  · | ·  ·  · | ·  ·  8 |C
+---------+---------+---------+
D| ·  8  · | 9  ·  3 | ·  7  · |D
E| ·  ·  · | ·  ·  · | 8  ·  5 |E
F| 7  ·  · | ·  ·  · | ·  9  · |F
+---------+---------+---------+
G| 9  ·  · | 6  ·  4 | ·  ·  · |G
H| ·  2  · | ·  1  · | ·  ·  · |H
J| ·  ·  6 | 7  ·  · | ·  ·  · |J
+---------+---------+---------+
1  2  3   4  5  6   7  8  9

These sudoku are not my own creations, they are permutations of other's work. I will credit their creators to the best of my ability when the puzzling is done.

• I find it interesting that all of the words from sudoku 1's sentence start with a letter that can be used for a coordinate. – Deusovi Jul 19 '16 at 7:07
• @Deusovi Heh, 13CIGARS8DOUGHNUT5AND2FALS3EXIT5FO2COLUMBO. – Jonathan Allan Jul 19 '16 at 7:52
• Is that comment a hint, or just something unintentional but mildly interesting? – Deusovi Jul 19 '16 at 7:53
• @Deusovi the latter, it was completely unintentional. – Jonathan Allan Jul 19 '16 at 8:00
• @ColdFrog the toughest are far less likely to be produced while maintaining Nikon's symmetric clues ideal, nice though it is. – Jonathan Allan Jul 19 '16 at 20:35

# Question 1

## This one was quite hard.

   1  2  3   4  5  6   7  8  9
+---------+---------+---------+
A| 1  2  6 | 4  9  7 | 8  3  5 |A
B| 9  5  4 | 8  6  3 | 1  2  7 |B
C| 8  7  3 | 5  2  1 | 9  6  4 |C
+---------+---------+---------+
D| 7  4  9 | 1  3  6 | 2  5  8 |D
E| 6  1  5 | 2  7  8 | 4  9  3 |E
F| 3  8  2 | 9  4  5 | 6  7  1 |F
+---------+---------+---------+
G| 2  3  7 | 6  8  4 | 5  1  9 |G
H| 4  9  1 | 3  5  2 | 7  8  6 |H
J| 5  6  8 | 7  1  9 | 3  4  2 |J
+---------+---------+---------+
1  2  3   4  5  6   7  8  9


# Question 2

## Easier than the first

   1  2  3   4  5  6   7  8  9
+---------+---------+---------+
A| 8  6  9 | 5  1  3 | 4  7  2 |A
B| 4  2  5 | 7  6  9 | 1  3  8 |B
C| 1  3  7 | 4  2  8 | 6  5  9 |C
+---------+---------+---------+
D| 7  8  2 | 3  9  4 | 5  6  1 |D
E| 5  9  4 | 1  8  6 | 3  2  7 |E
F| 3  1  6 | 2  5  7 | 9  8  4 |F
+---------+---------+---------+
G| 9  7  1 | 8  3  5 | 2  4  6 |G
H| 2  4  3 | 6  7  1 | 8  9  5 |H
J| 6  5  8 | 9  4  2 | 7  1  3 |J
+---------+---------+---------+
1  2  3   4  5  6   7  8  9


# Question 3

## This one took a while too

   1  2  3   4  5  6   7  8  9
+---------+---------+---------+
A| 3  1  2 | 8  9  6 | 5  4  7 |A
B| 8  7  9 | 4  2  5 | 1  3  6 |B
C| 6  5  4 | 1  3  7 | 9  2  8 |C
+---------+---------+---------+
D| 2  8  5 | 9  6  3 | 4  7  1 |D
E| 4  9  3 | 2  7  1 | 8  6  5 |E
F| 7  6  1 | 5  4  8 | 2  9  3 |F
+---------+---------+---------+
G| 9  3  8 | 6  5  4 | 7  1  2 |G
H| 5  2  7 | 3  1  9 | 6  8  4 |H
J| 1  4  6 | 7  8  2 | 3  5  9 |J
+---------+---------+---------+
1  2  3   4  5  6   7  8  9


All in all, it took me about a week in my free time to do all of these. I am on my holidays so I spent hours each day, racking my brain through these. Definitely some of the harder ones that I have seen.

## My Reasoning

I used the cross-hatching method as shown in this Link. And if that didn't work out, I would start over again.

• Welcome to Puzzling Stack Exchange. Note that the question says “and provide your reasoning”. – Peregrine Rook Aug 27 '16 at 3:58
• @PeregrineRook Noted and edited – Luciferangel Aug 27 '16 at 4:13
• Cross-hatching on its own would yield just one number across the three, a single 2 in 3. "22 premises for Sherlock." and then nothing more, so you did something beyond that; did you not take notes on other logical deductions you made? – Jonathan Allan Aug 27 '16 at 4:41
• @JonathanAllan Ahhh i see what you mean. I am not sure how to explain to you logically but yeah. i did it just with a pencil eraser and a lot of time. hahah. good questions tho – Luciferangel Aug 27 '16 at 9:56
• FYI my home brewed solver solves all of them in milliseconds :p It also gives no logical reasoning. – Jonathan Allan Aug 27 '16 at 9:59

I solved these Sudoku with a computer program. Its outputs ("reasonings") for each of the given boards can be found here.

My solutions are the same as Luciferangel's, so I won't repeat them here in the post (you can find them at the bottom of the files linked above).

My program performs a recursive backtracking search, which works as follows:

• The algorithm (deterministically) chooses a cell and makes a guess as to its value (e.g. guessing that cell A3 is a 3).

• Then it tries to solve the puzzle taking that guess as an assumption.

• If no solution is found (i.e. backtracking), it tries a different guess (e.g. guessing that cell A3 is a 6).

• When all possible values for that cell have been exhausted, the algorithm outputs 'no solution' (e.g. cell B2 cannot be filled)

I made a couple optimizations to better simulate how a human might approach the puzzle.

• For every cell, the program takes the intersection of the missing numbers in its corresponding row, column, and box. If this intersection has only one number, it immediately fills in that cell (e.g. cell A7 can only be a 8). If the intersection has no numbers in it, it outputs 'no solution' (e.g. cell H9 cannot be filled).

• For every row, column, and box, the program looks at the intersections computed above to see how many cells each missing number could go in to. If only one box could be filled with a given number, it immediately fills in that cell (e.g. cell B1 is the only cell in the top-left box that can be a 6). If no boxes can be filled with a given number, it outputs 'no solution' (e.g. no 1s can be placed in row G).

Each of the puzzles takes on the order of a thousand steps to solve (as opposed to my previous solution which took on the order of a hundred thousand steps per board).

• best explanation ever :) – Oray Aug 28 '16 at 8:57
• Could you maybe include an excerpt from one of them at the very least? This answer is next to useless on e.g., mobile without zip support. Even Pastebin or gist would be preferable. – Will Aug 28 '16 at 13:30
• My word they are long ...and pretty hard to follow (nesting might help). – Jonathan Allan Aug 28 '16 at 13:31
• So this is effectively pencil mark, guess and check, repeat? It does not look like it makes any reasoned decision about what to guess and check (just goes left-to right, top-to bottom). Code? – Jonathan Allan Aug 28 '16 at 13:36
• @Will I know, but unfortunately even one of the files by itself exceeds the 10 MB limit on a Gist, and if I put the files up unzipped the Drive previewer chokes on them. – 2012rcampion Aug 28 '16 at 18:42

I don't think you are going to get a better logical explanation of a solve for these. I ran them through Sudoku Explainer and they all got a score of SE 11.8 or SE 11.9 which is well in the range of not reasonably human solvable. I also ran them through multiple other human style solvers and they all choked. It's possible there is some obscure method that none of the solvers I used have implemented yet but it is more likely there is no straightforward human friendly logical solution to them. Following is the output of Sudoku Explainer for each board. If you want more step by step info I suggest you get Sudoku Explainer:

13 cigars, 8 doughnuts, and 2 false exits for Columbo.
Analysis results
Difficulty rating: 11.8
This Sudoku can be solved using the following logical methods:
55 x Hidden Single
1 x Direct Hidden Pair
1 x Naked Single
3 x Pointing
2 x Claiming
5 x Naked Pair
1 x Hidden Pair
1 x BUG type 1
5 x Turbot Fish
1 x Forcing X-Chain
2 x Bidirectional Cycle
5 x Forcing Chain
1 x Nishio Forcing Chains
1 x Cell Forcing Chains
4 x Region Forcing Chains
6 x Dynamic Cell Forcing Chains
15 x Dynamic Region Forcing Chains
8 x Dynamic Contradiction Forcing Chains
7 x Dynamic Contradiction Forcing Chains (+)
1 x Dynamic Double Forcing Chains (+)
9 x Dynamic Contradiction Forcing Chains (+ Forcing Chains)
6 x Dynamic Contradiction Forcing Chains (+ Multiple Forcing Chains)
1 x Dynamic Contradiction Forcing Chains (+ Dynamic Forcing Chains)

59 little grey cells for Poirot.
Analysis results
Difficulty rating: 11.8
This Sudoku can be solved using the following logical methods:
59 x Hidden Single
8 x Pointing
9 x Claiming
6 x Naked Pair
3 x Hidden Pair
4 x Naked Triplet
1 x Swordfish
1 x XYZ-Wing
3 x Bidirectional Y-Cycle
2 x Turbot Fish
1 x Forcing X-Chain
10 x Forcing Chain
1 x Bidirectional Cycle
4 x Nishio Forcing Chains
1 x Cell Forcing Chains
3 x Region Forcing Chains
1 x Dynamic Cell Forcing Chains
16 x Dynamic Contradiction Forcing Chains
1 x Dynamic Region Forcing Chains
5 x Dynamic Contradiction Forcing Chains (+)
2 x Dynamic Region Forcing Chains (+)
3 x Dynamic Contradiction Forcing Chains (+ Forcing Chains)
2 x Dynamic Contradiction Forcing Chains (+ Multiple Forcing Chains)
3 x Dynamic Cell Forcing Chains (+ Dynamic Forcing Chains)
1 x Dynamic Contradiction Forcing Chains (+ Dynamic Forcing Chains)
3 x Dynamic Region Forcing Chains (+ Dynamic Forcing Chains)

22 premises for Sherlock:
Analysis results
Difficulty rating: 11.9
This Sudoku can be solved using the following logical methods:
55 x Hidden Single
2 x Direct Hidden Pair
6 x Pointing
1 x Claiming
2 x Naked Pair
1 x Unique Rectangle type 1
1 x Turbot Fish
1 x Bidirectional Cycle
4 x Forcing Chain
2 x Cell Forcing Chains
4 x Region Forcing Chains
7 x Dynamic Region Forcing Chains
5 x Dynamic Contradiction Forcing Chains
1 x Dynamic Region Forcing Chains (+)
1 x Dynamic Contradiction Forcing Chains (+)
1 x Dynamic Region Forcing Chains (+ Forcing Chains)
6 x Dynamic Contradiction Forcing Chains (+ Forcing Chains)
6 x Dynamic Contradiction Forcing Chains (+ Multiple Forcing Chains)
6 x Dynamic Contradiction Forcing Chains (+ Dynamic Forcing Chains)
1 x Dynamic Cell Forcing Chains (+ Dynamic Forcing Chains)
2 x Dynamic Region Forcing Chains (+ Dynamic Forcing Chains)

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